I have sequence of squares: 1 -4 9 -16 .... Of course, it can be easily defined as: $a_{i} = (-1)^{i+1}*i^2$. I want to get generating function for this sequence and I know that it can be easily done for linear recurrent relation (ie when $a_i$ can be somehow linearly defined using previous members of sequence). But I failed to invent such relation for this sequence. How can it be done? Or how can we prove, that it is impossible? I'll be really grateful for any clues!
Sequence of squares is not recurrent
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sequences-and-series
discrete-mathematics
1 Answers
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$$a_1=1$$ $$a_2=-4$$ $$a_3=9$$ $$a_{n}=-3a_{n-1}-3a_{n-2}-a_{n-3}$$ for $n>3$
does the job
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0Huge thanks for your answer! What is the way to get such recurrent formulas? – 2017-02-28
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0The characteristic equation for a triple $-1$ is $(x+1)^3=x^3+3x^2+3x+1=0$. So, $c_1\cdot n^2\cdot (-1)^n$ is a solution of $a_n+3a_{n-1}+3a_{n-2}+a_{n-3}=0$. If you set $c_1=-1$ , you see that your expression is a solution. – 2017-02-28
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0Can you explain please a bit wider, how you use characteristic equation? How is it connected with the sequence? – 2017-02-28